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I am following a paper by Eberhard Hopf titled On S. Bernstein's Theorem on Surfaces z(x, y) of Nonpositive Curvature. Not long into the proof for his first lemma, he makes a construction of spheres that satisfy certain properties. Then, he mentions that these spheres are a monotonic family of surfaces.

I think I know what a family of surfaces is (ie, a set of surfaces). But what does he mean by monotonic? My guess would be that they do not intersect. But given the construction they always intersect in a circle. So perhaps my second guess would be that they only intersect on a plane.

EDIT: To be more specific; the spheres taken into account all intersect in a circle on the $z=0$ plane. Then he takes the part of the sphere which is above this plane and this collection of surfaces is the one that he calls a monotonic family of surfaces. So he's referring to surfaces which intersect at the border and nothing else.

EDIT 2 (probably should have done this sooner haha!): Here is the specific construction:

Suppose $z(x,y)$ is a function of class $C^2$ on a bounded open set $R\subset \mathbb{R^2}$. Now suppose that $z$ is also continuous on the border of $R$, which we call $B$. Now take $C$ to be a circle whose interior contains $R \cup B$.

Now we take the set $X = \{\text{spheres which intersect the }\: z=0\: \text{plane on }\: C\}$. From this set we construct now: $$ X^{+} = \{ \text{parts of the spheres from }X\: \text{that have}\: z\geq 0 \} $$ It is $X^{+}$ what he calls the monotonic family of surfaces.

D. Brito
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  • Maybe he's talking about a filtration? – ZxJx Nov 11 '19 at 16:50
  • What is a filtration? Pardon my ignorance – D. Brito Nov 11 '19 at 16:53
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    A filtration is a sequence of inclusions. This is probably not the case here, I spoke too soon! Can you give any more details? – ZxJx Nov 11 '19 at 16:59
  • @ZxJx sure! I can maybe give the full construction in the question. It is not long. – D. Brito Nov 11 '19 at 17:00
  • Ok, I think "monotonicity" refers to the distance between points in each surface. – ZxJx Nov 11 '19 at 17:14
  • I was wrong again! I was looking at Osserman's proof of Berstein's theorem, which I think he does differently. The monotonicity in your case is a ratio of areas, as in (https://pdfs.semanticscholar.org/11df/e420b35ba20c9fcfd27fd3f128157df63865.pdf), page 31, Theorem 2.6.2. I think one distiction to be made is that in the OP, there is no fixed center of these spheres. If you fix a center, then "monotonic" is referring to the monotonicity of a real function (the function takes the radius of a sphere to the ratio of areas) – ZxJx Nov 11 '19 at 17:52
  • In general "monotonic" refers to a function that preserves some partial order. So a "monotonic family of surfaces", one must specify two sets (in our case we want one set to be the parameter space with its endowed ordering) with respective partial orders, and a function between them. I think in this case, the spheres are parametrized by their radii, so we have a partial ordering on the spheres. In other situations, say if the parameter space has higher dimension, one would need to find a different partial order (often inclusion of sets) in order to speak of monotonicity. – ZxJx Nov 11 '19 at 18:06
  • Intuitively, if you take a ball and intersect it with your given surface, you can obtain a ratio of the area of the surface within the ball and the area of a disk inside the ball. The larger the ball, the larger this ratio will be. I think of this as giving the surface "more wiggle room", or from a different view: if your ball gets smaller, there is less "wiggle room" for the surface. – ZxJx Nov 11 '19 at 18:29
  • If you parametrize the spheres by the $z$-value of their centers, then there is a unique spherical slice in $X^+$ for each real number $z$. And indeed, every point in the upper half-space lies on exactly one of these slices. That is my guess for the interpretation. – Paul Sinclair Nov 12 '19 at 00:27
  • This is all too helpful. Thanks to both of you! – D. Brito Nov 12 '19 at 00:35

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